The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-archimedean (e.g., $p$-adic) settings. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely) many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right.
Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed. We prove also that the probl…
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of Tropical Differential Algebraic Geometry states that the support of solutions of systems of ordinary differential equations with formal power series coefficients over an uncountable algebraically closed field of characteristic zero can be obtained by solving a so-called tropicalized differential system. Tropicalized differential equations work on a completely d…
We show that solution sets of systems of tropical differential equations can be characterised in terms of monomial-freeness of an initial ideal. We discuss a candidate definition of tropical differential basis and give a nonexistence result for such bases in an example.
Given a complete real-valued field $k$ of residue characteristic zero, we study properties of a meromorphic $q$-differential form $ξ$ (a section of $Ω_X^{\otimes q}$) on a smooth proper $k$-analytic curve $X$. In particular, we associate to $(X,ξ)$ a natural tropical reduction datum combining tropical-geometric data over the value group of $|k^\times|$ and algebro-geometric reduction data over the residue field $\widetilde{k}$. We show that this datum satisfies natural compatibility conditions, and prove a lifting theorem asserting that any compatible tropical reduction datum lifts to an actua…
Let $I$ be an ideal of the ring of Laurent polynomials $K[x_1^{\pm1},\ldots,x_n^{\pm1}]$ with coefficients in a real-valued field $(K,v)$. The fundamental theorem of tropical algebraic geometry states the equality $\text{trop}(V(I))=V(\text{trop}(I))$ between the tropicalization $\text{trop}(V(I))$ of the closed subscheme $V(I)\subset (K^*)^n$ and the tropical variety $V(\text{trop}(I))$ associated to the tropicalization of the ideal $\text{trop}(I)$. In this work we prove an analogous result for a differential ideal $G$ of the ring of differential polynomials $K[[t]]\{x_1,\ldots,x_n\}$, where…
The Fundamental theorem of tropical differential algebra over nontrivially valued fields and the radius of convergence of nonarchimedean differential equations
Scuola Normale Superiore（高等师范学院）；CUNEF Universidad（CUNEF大学）
Elementary solutions of ordinary tropical differential equations, and vanishing orders of solutions of algebraic differential equations
Centro de Investigación en Matemáticas, A.C. (CIMAT)（墨西哥数学研究中心）；Universidad Autónoma Metropolitana-Azcapotzalco（墨西哥国立自治大学阿兹卡波察尔科分校）；Department of Physics and Mathematics, Universidad Iberoamericana Ciudad de México（墨西哥伊比利亚美洲大学物理与数学系）
Given a complete real-valued field $k$ of residue characteristic zero, we study properties of a differential form $ω$ on a smooth proper $k$-analytic curve $X$. In particular, we associate to $(X,ω)$ a natural tropical reduction datum combining tropical data of $(X,ω)$ and algebra-geometric reduction data over the residue field $\widetilde{k}$. We show that this datum satisfies natural compatibility condition, and prove a lifting theorem asserting that any compatible tropical reduction datum lifts to an actual pair $(X,ω)$. In particular, we obtain a short proof of the main result of a work [B…
Chambert-Loir and Ducros have recently introduced a theory of real valued differential forms and currents on Berkovich spaces. In analogy to the theory of forms with logarithmic singularities, we enlarge the space of differential forms by so called delta-forms on the non-archimedean analytification of an algebraic variety. This extension is based on an intersection theory for tropical cycles with smooth weights. We prove a generalization of the Poincaré-Lelong formula which allows us to represent the first Chern current of a formally metrized line bundle by a delta-form. We introduce the assoc…
We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer-Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich-Soibelman, Gross-Siebert and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer-Carta…
This is a survey article on ordinary differential equations over nonarchimedean fields based on the author's lecture at the 2015 Simons Symposium on nonarchimedean and tropical geometry. Topics include: the convergence polygon associated to a differential equation (or a connection on a curve); links to the formal classification of differential equations (Turrittin-Levelt); index formulas for de Rham cohomology of connections; ramification of finite morphisms; relations with the Oort lifting problem on automorphisms of curves. The appendices include some new technical results and an extensive t…
Chambert-Loir and Ducros have recently introduced real differential forms and currents on Berkovich spaces. In these notes, we survey this new theory and we will compare it with tropical algebraic geometry.
